A341709 -id:A341709 - cp's OEIS frontend

A341707 a(n) is the binary representation of n converted to yranib.

0, 1, 2, 1, 4, 3, 2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, -38, -39, -40, -41, -42, -43, -44, -45, -46, -47, -48, -49, -50, -51, -52, -53, 46, 45, 44, 43, 42, 41

Offset: 0

N. J. A. Sloane, Feb 18 2021

If n = Sum_{i=0..k} b_i*2^i, b_i = 0 or 1, b_k = 1, then a(n) = y(k) - Sum_{i=0..k-1} b_i*y(i), where y(j) = A004094(j) = 2^j written backwards in base 10.
If the 2^14 terms from a(16384) to a(32767) were to be considered a packet [call it #1], then the terms from a(32768) to a(49151) [call it #2] are #1 + 38362. #3 = #2 - 48361 (note that 48361 is the reverse of 16384). #4 = #3 + 25194. These successive displacements
     (38362, -48361,   25194, -48361,
     -38362, -48361,  341659, -48361,
     -38362, -48361,   71528, -48361,
     -38362, -48361,  369771, -48361,
     -38362, -48361,   71528, -48361,
     -38362, -48361,  -71491, -48361,
     -38362, -48361,   71528, -48361,
     -38362, -48361,  909934, -48361,
     -38362, -48361,   71528, -48361,
     -38362, -48361,  -71491, -48361,
     -38362, -48361,   71528, -48361,
     -38362, -48361,   27509, -48361,
     -38362, -48361,   71528, -48361,
     -38362, -48361,  -71491, -48361,
     -38362, -48361,   71528, -48361,
     -38362, -48361, 6786009, -48361,
     -38362, -48361,   71528, -48361,
     -38362, -48361,  -71491, -48361,
     -38362, -48361,   71528, -48361,
     -38362, -48361,   27509, -48361,
     -38362, -48361,   71528, -48361,
     -38362, -48361,  -71491, -48361,
     -38362, -48361,   71528, -48361,
     -38362, -48361,   27608, -48361,
     -38362, -48361,   71528, -48361, ...) fully describe the future of the sequence. Can we predict the values of the displacements from first principles? - Hans Havermann, Feb 24 2021

			If n = 48 = 110000_2, b_0 = ... = b_3 = 0, b_4 = b_5 = 1, so a(48) = A004094(5) - A004094(4) = 23 - 61 = -38, which is the first negative term (cf. A341708).

Eric Angelini, Posting to Math Fun Mailing List, Feb 18 2021

Jinyuan Wang, Table of n, a(n) for n = 0..10000

Cf. A004094.
See A341708 for the negative terms.
See A341709 for a different version of a yranib sequence.

{0}~Join~Array[Fold[Subtract, Reverse@ IntegerReverse[2^(-1 + Position[Reverse@ IntegerDigits[#, 2], 1][[All, 1]] )]] &, 69] (* Michael De Vlieger, Feb 25 2021 *)

/* Get decimal value of yranib representation of n written in binary (i.e., write n in binary, e.g., 9[10] = 1001[2], then read this in the yranib system,where the k-th position from the right has value s*R(2^k) where R=reverse(= decimal value read from right to left) and s = -1 except for the largest k. */
y2d(n)=if(n=binary(n),n[1]*=-1);-sum(k=0,#n-1,n[#n-k]*R(2^k))
R(n)=fromdigits(Vecrev(digits(n)))
apply(y2d, [0..99]) \\ M. F. Hasler, Feb 18 2021

def reverse(n):
    s = 0
    while n > 0:
        d, n = n%10, n//10
        s = 10*s+d
    return s
def A341707(n):
    s, t = 0, 1
    while n > 0:
        b, n = n%2, n//2
        if n > 0:
            s, t = reverse(t*b)+s, 2*t
        else:
            s = reverse(t*b)-s
    return s # A.H.M. Smeets, Feb 18 2021

Further terms from M. F. Hasler, Feb 18 2021

A341713 Indices of Ennesrem primes: k such that A004094(k)-1 is prime.

Original entry on oeis.org

2, 3, 13, 21, 347, 1217, 1267, 16459, 100909, 342243

Offset: 1

N. J. A. Sloane, Feb 20 2021

Numbers k such that reverse(2^k) - 1 is prime.

Is this sequence infinite?

			13 is a term, since 2^13 = 8192 -> 2918 -> 2917, which is prime.

N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.

Cf. A000043, A004094, A057708, A341709.

for(n=1,10000,my(pe=fromdigits(Vecrev(digits(2^n)))-1);if(ispseudoprime(pe),print1(n,", "))) \\ Hugo Pfoertner, Feb 20 2021

from sympy import isprime
def ok(k): return isprime(int(str(2**k)[::-1]) - 1)
for k in range(1, 2*10**3):
  if ok(k): print(k, end=", ") # Michael S. Branicky, Feb 20 2021

a(8) from Hugo Pfoertner, Feb 20 2021
a(9) from Hugo Pfoertner, Mar 22 2021
a(10) from Steven Charlton, Apr 26 2021

cp's OEIS Frontend

A341707 a(n) is the binary representation of n converted to yranib.

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